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{{MOS intro|Scale Signature=7L 3s}}
{{MOS intro|Scale Signature=7L 3s}}
Graham Breed has a  page on his website dedicated to 7+3 scales. He proposes calling the large step "t" for "tone", lowercase because the large step is a narrow neutral tone, and the small step "q" for "quartertone", because the small step is often close to a quartertone. (Note that the small step is not a quartertone in every instance of 7+3, so do not take that "q" literally.) Thus we have:
==Name==
TAMNAMS suggests the temperament-agnostic name '''dicoid''' (from dicot, an exotemperament) for the name of this scale.


t q t t t q t t q t
==Intervals==
==Names==
:''This article assumes [[TAMNAMS]] for naming step ratios, intervals, and scale degrees.''
This MOS is called '''dicoid''' (from [[dicot]], an exotemperament) in [[TAMNAMS]].
Names for this scale's [[degrees]], the positions of the scale's tones, are called '''mosdegrees'''. Its [[Interval|intervals]], the pitch difference between any two tones, are based on the number of large and small steps between the two tones and are thus called '''mossteps'''. Per TAMNAMS, both mosdegrees and mossteps are ''0-indexed'', and may be referred to as '''dicodegrees''' and '''dicosteps'''. Ordinal names, such as mos-1st for the unison, are discouraged for non-diatonic MOS scales.
{{MOS intervals|Scale Signature=7L 3s}}
Intervals of interest include:
 
* The '''perfect 3-mosstep''', or the scale's dark generator, whose range is around that of a neutral third. Its inversion, '''the perfect 7-mosstep''', has a range around that of a neutral sixth.
* The '''minor mosstep''', or '''small step''', which ranges form a quartertone to a minor second.
* The '''major mosstep''', or '''large step''', which ranges from a submajor second to a [[sinaic]] or trienthird (around 128¢).
* The '''major 4-mosstep''', whose range coincides with that of a perfect fourth; and its inversion, the '''minor 6-mosstep''', whose range coincides with that of a perfect 5th.
 
7L 3s combines the familiar sound of perfect fifths and fourths with the unfamiliar sounds of neutral intervals, thus making it compatible with Arabic and Turkish scales, but not with traditional Western scales.
 
== Theory ==
 
=== Temperament interpretations ===
 
=== Quartertone and tetrachordal analysis ===
Due to the presence of quartertone-like intervals, Graham Breed has proposed the terms ''tone'' (abbreviated as ''t'') and ''quartertone'' (abbreviated as ''q'') as alternatives for large and small steps. This interpretation is recommended for step ratios in which the small step approximates a quartertone.
 
Additionally, due to the presence of fourth and fifth-like intervals, 7L 3s can be analyzed as a [[tetrachord|tetrachordal scale]]. The perfect fourth can be traversed by 3 t's and a q, or 2 t's and a T.
 
I (Andrew Heathwaite]) offer "a" to refer to a step of 2t (for "augmented second")
 
Thus, the possible tetrachords are:


==Intervals==
T t t
The generator (g) will fall between 343 cents (2\7 - two degrees of [[7edo]] and 360 cents (3\10 - three degrees of [[10edo]]), hence a neutral third.
 
t T t


2g, then, will fall between 686 cents (4\7) and 720 cents (3\5), the range of [[5L 2s|diatonic]] fifths.
t t T


The "large step" will fall between 171 cents (1\7) and 120 cents (1\10), ranging from a submajor second to a [[sinaic]].
a q t


The "small step" will fall between 0 cents and 120 cents, sometimes sounding like a minor second, and sometimes sounding like a quartertone or smaller microtone.
a t q


The most frequent interval, then is the neutral third (and its inversion, the neutral sixth), followed by the perfect fourth and fifth. Thus, 7+3 combines the familiar sound of perfect fifths and fourths with the unfamiliar sounds of neutral intervals. They are compatible with Arabic and Turkish scales, but not with traditional Western ones.
t a q


Note: In TAMNAMS, a k-step interval class in dicoid may be called a "k-step", "k-mosstep", or "k-dicostep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.
t q a
{| class="wikitable"
 
!# generators up
q a t
!Notation (1/1 = 0)
 
!name
q t a
!In L's and s's
!# generators up
!Notation of 2/1 inverse
!name
!In L's and s's
|-
| colspan="8" style="text-align:center" | The 10-note MOS has the following intervals (from some root):
|-
|0
|0
|perfect unison
|0
|0
|0
|perfect 10-step
|7L+3s
|-
|1
|7
| perfect 7-step
|5L+2s
| -1
|3
|perfect 3-step
|2L+1s
|-
|2
|4
|major 4-step
|3L+1s
| -2
|6
|minor 6-step
|4L+2s
|-
|3
|1
|major (1-)step
|1L
| -3
|9v
|minor 9-step
|6L+3s
|-
|4
|8
|major 8-step
|6L+2s
| -4
|2v
|minor 2-step
|1L+1s
|-
|5
|5
|major 5-step
| 4L+1s
| -5
|5v
|minor 5-step
| 3L+2s
|-
|6
|2
|major 2-step
|2L
| -6
|8v
|minor 8-step
|5L+3s
|-
|7
|9
|major 9-step
| 7L+2s
| -7
|1v
|minor (1-)step
|1s
|-
|8
|6^
|major 6-step
|5L+1s
|  -8
|4v
|minor 4-step
|2L+2s
|-
|9
|3^
| augmented 3-step
| 3L
| -9
|7v
|diminished 7-step
|4L+3s
|-
|10
|0^
|augmented unison
|1L-1s
| -10
|0v
|diminished 10-step
|6L+4s
|-
|11
|7^
|augmented 7-step
|6L+1s
| -11
|3v
|diminished 3-step
|1L+2s
|-
| colspan="8" style="text-align:center" |The chromatic 17-note MOS (either [[7L 10s]], [[10L 7s]], or [[17edo]]) also has the following intervals (from some root):
|-
|12
|4^
| augmented 4-step
|4L
| -12
|6v
|diminished 6-step
|3L+3s
|-
|13
|1^
|augmented (1-)step
|2L-1s
| -13
|9w
|diminished 9-step
|5L+4s
|-
|14
|8^
|augmented 8-step
|8L+1s
| -14
|2w
|diminished 2-step
|2s
|-
|15
|5^
| augmented 5-step
|5L
| -15
|5w
|diminished 5-step
|2L+3s
|-
|16
|2^
|augmented 2-step
|3L-1s
| -16
|8w
|diminished 8-step
|4L+4s
|}
==Scale tree ==
==Scale tree ==
The generator range reflects two extremes: one where L = s (3\10), and another where s = 0 (2\7). Between these extremes, there is an infinite continuum of possible generator sizes. By taking freshman sums of the two edges (adding the numerators, then adding the denominators), we can fill in this continuum with compatible edos, increasing in number of tones as we continue filling in the in-betweens. Thus, the smallest in-between edo would be (3+2)\(10+7) = 5\17 – five degrees of [[17edo]]:
{{Scale tree|7L 3s}}
 
{| class="wikitable center-all"
{| class="wikitable center-all"
! colspan="6" rowspan="2" |Generator
! colspan="6" rowspan="2" |Generator
Line 267: Line 127:
|5\7||  ||  || || || ||857.143||342.857||1||0||→ inf||
|5\7||  ||  || || || ||857.143||342.857||1||0||→ inf||
|}
|}
TODO: add scale tree entries from old scale tree.


The scale produced by stacks of 5\17 is the [[17edo neutral scale]]. Between 11/38 and 16/55, with 9/31 in between, is the mohajira/mohaha/mohoho range, where mohaha and mohoho use the MOS as the chromatic scale of a [[Chromatic pairs|chromatic pair]].
The scale produced by stacks of 5\17 is the [[17edo neutral scale]]. Between 11/38 and 16/55, with 9/31 in between, is the mohajira/mohaha/mohoho range, where mohaha and mohoho use the MOS as the chromatic scale of a [[Chromatic pairs|chromatic pair]].
Line 273: Line 137:


You can also build this scale by stacking neutral thirds that are not members of edos – for instance, frequency ratios 11:9, 49:40, 27:22, 16:13 – or the square root of 3:2 (a bisected just perfect fifth).
You can also build this scale by stacking neutral thirds that are not members of edos – for instance, frequency ratios 11:9, 49:40, 27:22, 16:13 – or the square root of 3:2 (a bisected just perfect fifth).
==Rank-2 temperaments==
== 7-note subsets==
If you stop the chain at 7 tones, you have a heptatonic scale of the form [[3L 4s]]:
L s s L s L s
The large steps here consist of t+s of the 10-tone system, and the small step is the same as t. Graham proposes calling the large step here T for "tone," uppercase because it is a wider tone than t. Thus, we have:
T t t T t T t
This scale (and its rotations) is not the only possible heptatonic scale. Graham also gives us:
T t t T t t T
which is not a complete moment of symmetry scale in itself, but a subset of one.
==Tetrachordal structure==
Due to the frequency of perfect fourths and fifths in this scale, it can also be analyzed as a [[tetrachord|tetrachordal scale]]. The perfect fourth can be traversed by 3 t's and a q, or 2 t's and a T.
I (Andrew Heathwaite]) offer "a" to refer to a step of 2t (for "augmented second")
Thus, the possible tetrachords are:
T t t
t T t
t t T
a q t
a t q
t a q
t q a
q a t
q t a

Revision as of 05:57, 18 February 2024

This page is an in-progress rewrite for a main-namespace page. For the current page, see 7L 3s.

7L 3s, named dicoid in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 7 large steps and 3 small steps, repeating every octave. Generators that produce this scale range from 840 ¢ to 857.1 ¢, or from 342.9 ¢ to 360 ¢.

Name

TAMNAMS suggests the temperament-agnostic name dicoid (from dicot, an exotemperament) for the name of this scale.

Intervals

This article assumes TAMNAMS for naming step ratios, intervals, and scale degrees.

Names for this scale's degrees, the positions of the scale's tones, are called mosdegrees. Its intervals, the pitch difference between any two tones, are based on the number of large and small steps between the two tones and are thus called mossteps. Per TAMNAMS, both mosdegrees and mossteps are 0-indexed, and may be referred to as dicodegrees and dicosteps. Ordinal names, such as mos-1st for the unison, are discouraged for non-diatonic MOS scales.

Intervals of 7L 3s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-dicostep Perfect 0-dicostep P0dis 0 0.0 ¢
1-dicostep Minor 1-dicostep m1dis s 0.0 ¢ to 120.0 ¢
Major 1-dicostep M1dis L 120.0 ¢ to 171.4 ¢
2-dicostep Minor 2-dicostep m2dis L + s 171.4 ¢ to 240.0 ¢
Major 2-dicostep M2dis 2L 240.0 ¢ to 342.9 ¢
3-dicostep Perfect 3-dicostep P3dis 2L + s 342.9 ¢ to 360.0 ¢
Augmented 3-dicostep A3dis 3L 360.0 ¢ to 514.3 ¢
4-dicostep Minor 4-dicostep m4dis 2L + 2s 342.9 ¢ to 480.0 ¢
Major 4-dicostep M4dis 3L + s 480.0 ¢ to 514.3 ¢
5-dicostep Minor 5-dicostep m5dis 3L + 2s 514.3 ¢ to 600.0 ¢
Major 5-dicostep M5dis 4L + s 600.0 ¢ to 685.7 ¢
6-dicostep Minor 6-dicostep m6dis 4L + 2s 685.7 ¢ to 720.0 ¢
Major 6-dicostep M6dis 5L + s 720.0 ¢ to 857.1 ¢
7-dicostep Diminished 7-dicostep d7dis 4L + 3s 685.7 ¢ to 840.0 ¢
Perfect 7-dicostep P7dis 5L + 2s 840.0 ¢ to 857.1 ¢
8-dicostep Minor 8-dicostep m8dis 5L + 3s 857.1 ¢ to 960.0 ¢
Major 8-dicostep M8dis 6L + 2s 960.0 ¢ to 1028.6 ¢
9-dicostep Minor 9-dicostep m9dis 6L + 3s 1028.6 ¢ to 1080.0 ¢
Major 9-dicostep M9dis 7L + 2s 1080.0 ¢ to 1200.0 ¢
10-dicostep Perfect 10-dicostep P10dis 7L + 3s 1200.0 ¢

Intervals of interest include:

  • The perfect 3-mosstep, or the scale's dark generator, whose range is around that of a neutral third. Its inversion, the perfect 7-mosstep, has a range around that of a neutral sixth.
  • The minor mosstep, or small step, which ranges form a quartertone to a minor second.
  • The major mosstep, or large step, which ranges from a submajor second to a sinaic or trienthird (around 128¢).
  • The major 4-mosstep, whose range coincides with that of a perfect fourth; and its inversion, the minor 6-mosstep, whose range coincides with that of a perfect 5th.

7L 3s combines the familiar sound of perfect fifths and fourths with the unfamiliar sounds of neutral intervals, thus making it compatible with Arabic and Turkish scales, but not with traditional Western scales.

Theory

Temperament interpretations

Quartertone and tetrachordal analysis

Due to the presence of quartertone-like intervals, Graham Breed has proposed the terms tone (abbreviated as t) and quartertone (abbreviated as q) as alternatives for large and small steps. This interpretation is recommended for step ratios in which the small step approximates a quartertone.

Additionally, due to the presence of fourth and fifth-like intervals, 7L 3s can be analyzed as a tetrachordal scale. The perfect fourth can be traversed by 3 t's and a q, or 2 t's and a T.

I (Andrew Heathwaite]) offer "a" to refer to a step of 2t (for "augmented second")

Thus, the possible tetrachords are:

T t t

t T t

t t T

a q t

a t q

t a q

t q a

q a t

q t a

Scale tree

Template:Scale tree

Generator Cents L s L/s Comments
Chroma-positive Chroma-negative
7\10 840.000 360.000 1 1 1.000
40\57 842.105 357.895 6 5 1.200 Restles↑
33\47 842.553 357.447 5 4 1.250
59\84 842.857 357.143 9 7 1.286
26\37 843.243 356.757 4 3 1.333
71\101 843.564 356.436 11 8 1.375
45\64 843.750 356.250 7 5 1.400 Beatles
64\91 843.956 356.044 10 7 1.428
19\27 844.444 355.556 3 2 1.500 L/s = 3/2, suhajira/ringo
69\98 844.698 355.102 11 7 1.571
50\71 845.070 354.930 8 5 1.600
81\115 845.217 354.783 13 8 1.625 Golden suhajira
31\44 845.455 354.545 5 3 1.667
74\105 845.714 354.286 12 7 1.714
43\61 845.902 354.098 7 4 1.750
55\78 846.154 353.846 9 5 1.800
12\17 847.059 352.941 2 1 2.000 Basic dicoid
(Generators smaller than this are proper)
53\75 848.000 352.000 9 4 2.250
41\58 848.273 351.724 7 3 2.333
70\99 848.485 351.515 12 5 2.400 Hemif/hemififths
29\41 848.780 351.220 5 2 2.500 Mohaha/neutrominant
75\106 849.057 350.943 13 5 2.600 Hemif/salsa/karadeniz
46\65 849.231 350.769 8 3 2.667 Mohaha/mohamaq
63\89 849.438 350.562 11 4 2.750
17\24 850.000 350.000 3 1 3.000 L/s = 3/1
56\79 850.633 349.367 10 3 3.333
39\55 850.909 349.091 7 2 3.500
61\86 851.163 348.837 11 3 3.667
22\31 851.613 348.387 4 1 4.000 Mohaha/migration/mohajira
49\69 852.174 347.826 9 2 4.500
27\38 852.632 347.368 5 1 5.000
32\45 853.333 346.667 6 1 6.000 Mohaha/ptolemy
5\7 857.143 342.857 1 0 → inf

TODO: add scale tree entries from old scale tree.


The scale produced by stacks of 5\17 is the 17edo neutral scale. Between 11/38 and 16/55, with 9/31 in between, is the mohajira/mohaha/mohoho range, where mohaha and mohoho use the MOS as the chromatic scale of a chromatic pair.

Other compatible edos include: 37edo, 27edo, 44edo, 41edo, 24edo, 31edo.

You can also build this scale by stacking neutral thirds that are not members of edos – for instance, frequency ratios 11:9, 49:40, 27:22, 16:13 – or the square root of 3:2 (a bisected just perfect fifth).

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